Question

In Triangle MNK MN=NK, the measure of angle N is 110°, and MK=5. What is the length of MN?

Please help

Answer 1

Explanation:

Since MN = NK, we know that triangle MNK is an isosceles triangle, and therefore angles MKN and MNK are congruent. Let x be the measure of angles MKN and MNK, in degrees. Then, we have:

x + x + 110° = 180° (sum of angles in a triangle)

2x = 70°

x = 35°

Now, using the Law of Cosines, we can find the length of MN. Let y be the length of MN. Then, we have:

MK² + NK² - 2(MK)(NK)cos(N) = MN²

Substituting the given values, we get:

5² + y² - 2(5)(y)cos(110°) = y²

Simplifying, we get:

25 + y² + 10y(cos 110°) = y²

cos 110° is negative, so we have:

25 + 10y(cos 110°) = 0

10y(cos 110°) = -25

y(cos 110°) = -2.5

y = (-2.5) / cos 110°

Using a calculator, we get:

y ≈ 12.16

Therefore, the length of MN is approximately 12.16.